First-order Cohomology Groups Of Non-ordered Witt Algebras

Cohomology of infinite dimensional Lie algebras have drawn more and more attentions be-cause of their great importance and applications.As generalized symmetries in theoretical physics,non-graded Witt algebra W was found by Senyue Lou in 1993[14].The structure of non-graded Witt algebra is more complicated than that of Witt algebra,which makes it much more difficult and more challenging to investigate.In this paper,we determine the first cohomology group of non-graded Witt algebra W with coefficients in its tensor density module Fa’,a,b.Our main result is the following theorem where the linear operators Di（i=1,2,3,4,5,6）are defined as the following:D0(L_α,m)=αV_α,m+mV_α,m-1,D1(L_α,m)=α²V_α,m+2mαV_α,m-1+m（m-1）V_α,m-32,D2(L_α,m)=α³V_α,m+3mα²V_α,m-1,+3αm（m-1）V_α,m-32+m（m-1）（m-2）V_α,m-33,for all α,m ∈ Z.